Where the whale-things are?

David L Miller (@millerdl)

University of Melbourne

15 April 2016

Slides available at http://converged.yt

• Currently:
  • NOAA Northeast Fisheries Science Center (sic), Woods Hole, MA
  • (“Honourary” (ha!) position at St Andrews)
• Previously:
  • Centre for Research into Ecological and Environmental Modelling, St Andrews
  • University of Rhode Island
  • PhD Bath, UK; spatial modelling w/ Simon Wood
• Why am I here?
  • Collab w/ Mark Bravington CSIRO, Nat Kelly Antarctic Division

(a talk)

• Want to know where they are
  • whale watching/stock assessments
• Want to know where they aren't
  • Navy/seismic/oil/shipping/fishing etc
• Want to know what they like/dislike
  • ecology/biology

• “Fit to the histogram”
• Model:

\[ \mathbb{P} \left[ \text{animal detected } \vert \text{ animal at distance } y\right] = g(y;\boldsymbol{\theta}) \]

• Calculate the average probability of detection:

\[ \hat{p} = \frac{1}{w} \int_0^w g(y; \boldsymbol{\hat{\theta}}) \text{d}y \]

• Covariates that affect detectability (Marques et al, 2007)
• Perception bias (\( g(0)<1 \)) (Burt et al, 2014)
• Availability bias (Winiarski et al, 2013; Borchers et al, 2013)
• Detection function formulations (Miller and Thomas, 2015)
• Measurement error (Marques, 2004)  
  

Figure from Marques et al (2007)

• Rescale the (flat) density and extrapolate

\[ \hat{N} = \frac{\text{study area}}{\text{survey area}}\sum_{i=1}^n \frac{s_i}{\hat{p}_i} \]

(where \( s_i \) are group/cluster sizes)

Rather than hoping our design detects to the changes in density and contriving a complex set of strata, use a model!

• a priori, no idea how “bad” your design is \( \Rightarrow \) use a model.
• a posteriori, no idea how “bad” your design was \( \Rightarrow \) use a model.

Hedley and Buckland (2004)

Miller et al. (2013)

Taking the previous example…

\[ n_j = \color{red}{A_j}\color{blue}{\hat{p}_j} \color{green}{\exp}\left[\color{grey}{ \beta_0 + \sum_k f_k(z_{kj})} \right] + \epsilon_j \]

where \( \epsilon_j \sim N(0, \sigma^2) \), \( \quad n_j\sim \) count distribution

• \( \color{red}{\text{area of segment}} \)
• \( \color{blue}{\text{probability of detection in segment}} \)
• \( \color{green}{\text{(inverse) link function}} \)
• \( \color{grey}{\text{model terms}} \)

library(dsm)
dsm_env_tw <- dsm(count~s(Depth) + s(NPP) + s(SST),
                  ddf.obj=df_hr,
                  segment.data=segs, observation.data=obs,
                  family=tw(), method="REML")

(method="REML" uses REML to select wigglyness; Wood, 2011)

dsm is based on mgcv by Simon Wood

• Need to make smooths “wiggly enough”
• Term selection
  • shrinkage (Marra & Wood, 2011)
  • \( p \) values (Marra & Wood, 2012)
• Residuals check plots (Dunn and Smyth, 1996)

“Everything is related to everything else, but near things are more related than distant things”

Tobler (1970)

• Careful inclusion of terms
• Fit models using robust criteria (REML)
• Test for concurvity (mgcv::concurvity)
• Test for sensitivity (lots of models)

• Need to propagate uncertainty!
• This is hard, but can be done!
• Refit model with “extra” term

\[ n_j = A_j p_j \exp \left\{ \color{red}{\left[\frac{ \partial \log p_j(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}} \Big\vert_{\boldsymbol{\theta} = \hat{\boldsymbol{\theta}}}\right] \boldsymbol{\gamma}} + \beta_0 + \sum_k f_k(z_{jk}) \right\} \]

• random effect – fix the corresponding variance matrix \( \boldsymbol{\gamma} \sim N(0,-\mathbf{H}^{-1}_\theta) \)
• Bravington, Hedley & Miller (in prep)

• Detectability \( \Rightarrow \) effective area surveyed (offset in model)
• Multi-stage models are handy!
• Understand (and check) each part
• Split your modelling efforts amongst people

• This methodology is general
  • Bears, birds, beer cans, Loch Ness monsters can all be modelled
• Models are flexible!
  • Linear things, smooth things, random effect things (and more)
• If you know GLMs, you can get started with DSMs
  • Mature theoretical basis, still lots to do
• Active user community, active software development

• St Andrews: Eric Rexstad, Len Thomas, Louise Burt, Laura Marshall, Steve Buckland, Sharon Hedley
• Duke: Jason Roberts, Laura Mannocci
• CSIRO: Mark Bravington
• Antarctic Division: Natalie Kelly
• Funding: University of St Andrews, State of Rhode Island, US Navy, NOAA, International Whaling Commission

Slides w/ references available at converged.yt

• Integration of derivative (squared) gives wigglyness
• Fit needs to be penalised
• Penalty matrix gives the wigglyness
• Estimate the \( \beta_k \) terms but penalise objective
  • “closeness to data” + penalty

• For each \( b_k \) calculate the penalty
• Penalty is a function of \( \beta \)
  • \( \lambda \beta^\text{T}S\beta \)
• \( S \) calculated once
• smoothing parameter (\( \lambda \)) dictates influence

• We can set basis complexity or “size” (\( k \))
  • Maximum wigglyness
• Smooths have effective degrees of freedom (EDF)
• EDF < \( k \)
• Set \( k \) “large enough”